Help with basic seven segment display

[Nerfhurter]

Hello, I'm fairly new to electrical engineering, in my class we are working with 7 segment display (common anode).

I've come up with the following truth table for "D" input (~ is NOT).

           ~Z      Z

~X~Y    0       0  

~X  Y    1       1

  X  Y    1       0

  X~Y    1       1

I used k-maps to get the following simplified expression Hi=~XY+X~Z+X~Y. For the third term I use an invert after ~XY, but that makes one expression ORd with its inverse. This is giving, as expected 1's for all possible inputs. I'm wondering where I went wrong, any help is much appreciated.

[Nerfhurter]

Bump: I understand that because of boolean simplifying theorems an expression summed with its inverse is just 1, but then where am I going wrong?

[Jordan Cole]

Im not sure exactly what your asking up there but heres a SSD wired to seven switches that might be useful for figuring out what LED's need to be on to produce whatever output your looking for. (Its not entirely clean so if you want a clean one tell me and ill upload another one. It will be saved as .ms11)